Archimedes calculated pi by using the perimeter of inscribed and circumscribed polygons to create upper and lower bounds for the circumference of a circle.
Archimedes' Method for Approximating Pi
Archimedes understood that pi (π) represents the ratio of a circle's circumference to its diameter. Lacking modern computational tools, he devised a clever geometric method to approximate this value. His approach involved the following key steps:
1. Inscribed and Circumscribed Polygons
Archimedes began by drawing regular polygons inside (inscribed) and outside (circumscribed) a circle. The perimeters of these polygons provided estimates of the circle's circumference.
2. Perimeter as Approximation
- Inscribed Polygon: The perimeter of the inscribed polygon is less than the circumference of the circle. This gives a lower bound for π.
- Circumscribed Polygon: The perimeter of the circumscribed polygon is greater than the circumference of the circle. This gives an upper bound for π.
3. Increasing the Number of Sides
Archimedes recognized that the more sides the polygons had, the closer their perimeters would be to the actual circumference of the circle, and therefore, the more accurate his approximation of π would become. He started with hexagons (6 sides) and progressively doubled the number of sides to 12, 24, 48, and finally, 96-sided polygons.
4. Calculating the Perimeters
The challenging part was precisely calculating the perimeters of these polygons. Archimedes used geometric relationships (based on the Pythagorean theorem and trigonometric identities derived from geometric proofs) to iteratively determine the side lengths of the polygons as he doubled their number of sides. This required careful geometric reasoning and meticulous calculations.
5. Deriving Bounds for Pi
By calculating the perimeters of the inscribed and circumscribed 96-sided polygons, Archimedes obtained the following bounds:
- Lower bound: 3 10/71
- Upper bound: 3 1/7
This means Archimedes showed that π lies between 3.1408 and 3.1429, a remarkably accurate approximation for his time. His result is often expressed as:
3 10/71 < π < 3 1/7
Table: Illustration of Archimedes' Approach
| Polygon Type | Number of Sides | Perimeter Calculation (Conceptual) | Implication for Pi Calculation |
|---|---|---|---|
| Inscribed | Increasing | Use geometric relationships to calculate the perimeter inside the circle | Provides a lower bound for the value of Pi |
| Circumscribed | Increasing | Use geometric relationships to calculate the perimeter outside the circle | Provides an upper bound for the value of Pi |
Significance of Archimedes' Method
Archimedes' method was groundbreaking because it:
- Provided a rigorous way to approximate an irrational number like π.
- Demonstrated the power of geometric reasoning and iterative calculations.
- Established a foundation for later mathematicians to refine the value of π using similar techniques.
